3 Answers

  1. Before formulating the Liar's Paradox so that the reader does not feel deceived, it is necessary to agree on terms. Assume that any statement can be either true or false. This assumption looks plausible for simple statements: “A pigeon is a bird”, “Vasya has a cat at home”. And for more complex statements, our assumption also looks plausible: “Vasya has a cat or dog at home”, “All swans are white”. But, as the statements become more complex, we will have more doubts that our assumption is correct: “I knew who would become the 45th president of the United States before the election” – can we determine whether this assumption is true or false?

    The Liar's paradox is a simple way to decide the fate of our assumption once and for all. “This statement is false” – is this statement true or false? If our assumption is correct, then one of the two options must be fulfilled.

    Let's say that this statement is true, so what is claimed is true, so this statement is false. This is a contradiction, because we assumed that any statement can be either true or false, but not both at the same time.

    Let's say that our statement is false, so what is being said is not true, so this statement is not false. Once again, our assumption runs into a contradiction.�

    The idea that there are statements whose truth cannot be determined is counterintuitive. This contradiction creates conflicts in everyday life and moves science forward.�

    Reflecting on this contradiction, we can easily come to doubt the knowability of the world. Looking at the progress of science, you might think that it is enough to accurately measure everything and write out all the laws of nature in order to definitely answer any question about the past and future from the most pressing ones, like ” Who ate all the plums?” to the most general ones: “what is the age of the universe?”. At the beginning of the 20th century, science received two important results, thanks to which we can stop worrying about the knowability of the world. The first result , the Heisenberg uncertainty Principle, showed that there is a positive limit to the accuracy of measurements. The second result, Godel's Incompleteness Theorem, showed that there are statements in logical theories whose truth cannot be verified within the framework of the theory itself. Moreover, Godel's theorem is not about some exotic examples of logical theories, but about any theory in which addition and multiplication of natural numbers can be expressed.

    Another important paradox, very similar to the Liar's Paradox, showed that set theory is not just an exercise in rewriting formulas, but an important branch of mathematics. This paradox goes like this: “If the elements of a certain set are all sets that do not contain themselves as an element, then does this set contain itself?” Just as in the case of the liar's paradox, we can look at two options and make sure that both options lead to a contradiction. Thus, we conclude that such a set cannot exist. The knowledge that not all sets that can be defined exist allowed mathematicians to develop a whole new direction in their science, the essence of which is to search for a language that can be used to define only those sets that can exist.

    This version of the Liar's Paradox was first published by the British scientist Bertrand Russell in the book Principles of Mathematics in 1903.

  2. The liar's paradox is a statement like ” I'm not telling the truth.” If this statement is true, then it is false, and vice versa. This cannot be the case according to the first law of logic, namely, that the subject under discussion is only itself; the misunderstanding arises from an apparent choice between truth and falsehood, which is not the case here.�

    The words “I lie” are neither true nor false without a previous statement to which they might refer. If you do not substitute concepts, the paradox disappears and nonsense remains so.

  3. An example of the liar's paradox can be a statement like ” Everything you read is a lie.” If this is true, then the statement itself turns out to be true. But if everything you read is really a lie, then the very statement you read is false – a vicious circle. That is, the liar's paradox is a statement for which it is impossible to say exactly whether it is true or false.

    One of the first statements like this paradox is attributed to Epimenides (VII century BC). He was a Cretan and said:“All Cretans are liars.”� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � So is the phrase ” Pinocchio “.I'm about to grow a nose”� � � � � � � � � � � � � � � � � � � � � � �You can imagine a situation where a person is slapped in the face for making a false statement and asked: “What is it?”Are we going to hit you now?“. Even if you didn't plan to hit them, and the person says “yes”, they will get a punch in the face.

    There is a solution to this paradox: a statement should simply be considered with non-strict conditions, for example, ” I lie, but not always. “

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